Automorphism Group of the Incidence Structure and Its Orbits

Dedicated to the memory of Dean Neumann.

“When you’re the smartest guy in the room, you don’t need to say so.”

0Notation

The plain-English pages and cards name accounts with letters: TB, LDC, and A to K for the eleven smaller accounts of the group posts, lettered by first appearance. In group theory single capital letters already have jobs (a subgroup, the group, a block, a set), so this paper writes the accounts as points \(p_1,\dots,p_{13}\).

So the pair the plain pages call TB, LDC is \(\{p_1,p_2\}\), and the trio that appeared once, together, on April 10 is \(\{p_{10},p_{11},p_{12}\}\). Elsewhere: \(B_1,\dots,B_7\) are the seven group posts in date order; \(G=\operatorname{Aut}(\mathcal S)\); \(C\) is the centre of a pendant star (Lemma 3.1); \(r_x\) and \(\lambda_{xy}\) are replication and pair counts. In the 492-post structure the other 159 points carry IDs from the published data (F001–F068, S001–S099), with \(p_1\) = timballard89 throughout.

1Setting

The Levi (incidence) graph \(L(\mathcal S)\) is bipartite on \(P\sqcup\mathcal B\), with \(x\sim B\) iff \(x\in B\). Colour-preserving automorphisms of \(L\), restricted to \(P\), are exactly \(\operatorname{Aut}(\mathcal S)\); this is how Sage computes it (canonical labelling of a coloured graph).

2The seven group posts

So \(v=13\), \(b=7\), \(k=5\), with replication numbers \(r_{p_{1}}=r_{p_{2}}=7\), \(r_{p_{3}}=r_{p_{6}}=4\), \(r_{p_{4}}=3\), \(r_{p_{5}}=r_{p_{9}}=2\), and \(r=1\) for \(p_{7}\), \(p_{8}\), \(p_{10}\), \(p_{11}\), \(p_{12}\) and \(p_{13}\). Only 40 of the \(\binom{13}{2}=78\) pairs have \(\lambda_{xy}>0\); since \(b\lt v\), Fisher’s inequality rules out any 2-design structure, and Sage confirms \(\mathcal S_7\) is only a trivial \(0\)-design. The Gram matrix is \(NN^{\mathsf T}=\operatorname{diag}(r)+\Lambda\), and \(NN^{\mathsf T}=\sum_{B}\mathbf 1_B\mathbf 1_B^{\mathsf T}\): each block contributes a \(K_5\) to the pair graph.

Orbit-stabilizer. With \(G=\operatorname{Aut}(\mathcal S_7)\), \(|G|=|\operatorname{Orb}(x)|\,|\operatorname{Stab}(x)|\) for every point (Gallian, Ch. 7):

Burnside. The number of orbits is \(\frac1{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|\). By cycle type on \(\{p_{1},p_{2}\}\times\{p_{10},p_{11},p_{12}\}\):

So there are \(120/12=10\) orbits: \(\{p_{1},p_{2}\}\), \(\{p_{10},p_{11},p_{12}\}\) and the eight singletons \(p_{3}\), \(p_{4}\), \(p_{5}\), \(p_{6}\), \(p_{7}\), \(p_{8}\), \(p_{9}\) and \(p_{13}\). Sage’s fixed-point counts over \(G\) are exactly \([8,8,9,9,9,10,10,11,11,11,11,13]\). The six element types are the six conjugacy classes of \(D_6\).

Reading: in the group-post format \(p_{1}\) and \(p_{2}\) are indistinguishable within this structure, and \(\{p_{10},p_{11},p_{12}\}\) was a batch swapped in whole. What separates \(p_{1}\) from \(p_{2}\) lies outside \(\mathcal S_7\): \(p_{1}\)’s own post crediting \(p_{2}\) on December 20, 2025.

3All 492 collaborations

Let \(\mathcal S_{492}\) have as blocks every captured post on which \(p_{1}\) and at least one other account appear (492 posts; 74 caption-only mentions excluded), on \(v=172\) points, block sizes 2 to 5. Sage finds \(|G|=3{,}343{,}786{,}911{,}907{,}539{,}166{,}867{,}488{,}768{,}000=2^{60}3^{16}5^{3}7^{2}11\) and 101 orbits: 73 fixed points and 28 nontrivial orbits containing 99 points. There are 33 twin classes (25 of size 2, six of size 3, two of size 4).

Remark. One of us has called Lemma 3.1 the “pedant-star lemon” for years without ever having an example of it. Here are four at once, all centred on \(p_{1}\).

Primitivity as reported by Sage’s is_primitive(); in the imprimitive cases the block system is the set of twin classes.

Burnside, factor by factor. Summing \(|\operatorname{Fix}(g)|\) over \(3.3\times10^{30}\) elements is out of the question, but Theorem 3.2 makes Burnside’s lemma easy. For a direct product \(G=\prod_O G^O\) acting on the disjoint union of the \(O\), \(\operatorname{Fix}(g)\) splits over the factors, so \[\frac1{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|=\sum_{O}\frac1{|G^O|}\sum_{h\in G^O}|\operatorname{Fix}_O(h)|=\sum_O\#\{\text{orbits of }G^O\text{ on }O\}.\] Each \(G^O\) is transitive on its \(O\) (one orbit each, 28 in all), and each of the 73 fixed points is its own orbit: \(28+73=101\), matching Sage. On \(\mathcal S_7\) the same formula gives \(2+8=10\), matching the 120/12 computed by hand above.

The cross-check. \(\{p_{10},p_{11},p_{12}\}\) reappears as one of the three \(S_3\) orbits of \(\mathcal S_{492}\), found with no reference to \(\mathcal S_7\).

Size. The pendant-star factors dominate: \(|S_2\wr S_7|\cdot|S_{11}|\cdot|S_3\wr S_3|\cdot|S_4\wr S_2|\approx3.8\times10^{19}\) before the smaller factors multiply in. Sage finds the orbits from the generators (Schreier–Sims), not by enumerating \(G\).

Reading: 73 accounts are fixed points, the anchors the structure depends on (\(p_{1}\) on all 492 blocks, his foundation on 223, tbfrescue on 179, his film on 24, \(p_{2}\) on 8). The large orbits are pendant stars: posts made of \(p_{1}\) and one-off guests, whose guests are interchangeable both within a post (twins) and across posts (the top group of the wreath product).

4Sage session

B = [["p1","p2","p3","p4","p5"], ["p1","p2","p3","p5","p6"], ["p1","p2","p3","p7","p6"],
     ["p1","p2","p3","p4","p6"], ["p1","p2","p8","p4","p9"], ["p1","p2","p10","p11","p12"],
     ["p1","p2","p13","p9","p6"]]
S = IncidenceStructure(B); G = S.automorphism_group()
G.order(), G.structure_description(), G.gens()
#  (12, 'D6', (('p1','p2'), ('p10','p11'), ('p12','p10')))
G.orbits()
#  (('p9',), ('p8',), ('p12','p11','p10'), ('p7',), ('p3',), ('p1','p2'), ('p13',), ('p6',), ('p4',), ('p5',))
G.stabilizer('p10').structure_description()        #  'C2 x C2'
sum(len([p for p in S.ground_set() if g(p) == p]) for g in G) / G.order()   #  10
S.is_t_design(return_parameters=True)             #  (True, (0, 13, 5, 7))
L = S.incidence_graph(); L.order(), L.size()       #  (20, 35)

SageMath 10.9. The 492-post blocks are published as posts.csv (smaller accounts as IDs); the same calls on them give Theorem 3.2.

5Results and replication

Results. Every claim above was computed more than once, by independent methods, before and after the fact.

Replication of the data. The mathematics is only as good as the incidence matrix. Anyone can rebuild \(N\) for \(\mathcal S_7\) from Instagram itself:

1. open the seven posts (links in group_posts.csv);
2. record each post’s uploader and its Collaborators list;
3. run build_matrix.py and then verify_automorphisms.py.

The test needs no knowledge of our labels, because \(|\operatorname{Aut}(\mathcal S)|\) and the orbit sizes are invariant under relabelling the points. The step-by-step procedure and pass/fail table are in TEST.md beside the data.

Evidence package. The posts as we hold them (\(B_1\)–\(B_6\); \(B_7\) was deleted from Instagram between Oct 5 and Oct 13, 2026, so what survives of it is released as dated capture records rather than media) are released in evidence/, with a manifest:

• each watermarked file, with the block it belongs to (\(B_1\)–\(B_6\));
• the post link;
• Instagram’s upload time and uploader;
• SHA-256 hashes of both the original and the watermarked file.

6Scope

The structure is only as complete as the captures: \(p_{1}\)’s grid back to September 11, 2025; some accounts capped; captions truncated (which affects only the excluded mention-only posts). “Same orbit” means identical position in this incidence structure, not common control. The proofs are machine-checked (SageMath 10.9, networkx) and the results were reviewed by two AI systems, Claude and Grok; the paper has not yet been refereed by mathematicians. Corrections are welcome.

7Acknowledgements

With thanks to Dave Meronk, for explaining Gallian’s text; Charles Holland, for algebra; Charles Ford, for representation theory; Jay Goldman, for combinatorics; and Stephen H. McCleary, on telling left from right [4]. Results reviewed by Claude (Anthropic, @claude) and Grok (xAI, @grok), October 2026.

8References

1. J. A. Gallian, Contemporary Abstract Algebra, Chapman & Hall/CRC (10th ed. 2021; 11th ed.). Ch. 7: orbits, stabilizers, the orbit-stabilizer theorem; “Symmetry and Counting”: Burnside’s lemma.
2. J. D. Dixon and B. Mortimer, Permutation Groups, Graduate Texts in Mathematics 163, Springer, 1996 (wreath products, primitivity).
3. B. D. McKay and A. Piperno, “Practical graph isomorphism, II,” J. Symbolic Comput. 60 (2014) 94–112 (nauty/Traces).
4. S. H. McCleary, “Telling Left from Right,” in Ordered Algebraic Structures, Springer (Kluwer), 1989, doi:10.1007/978-94-009-2472-7_7.
5. The SageMath Developers, SageMath 10.9: IncidenceStructure, automorphism_group.
6. Companion pages (all pages): Who went first, The Name-Tag Test, Sun, Moon and Starlink; related: B₇ and the Lettering of the Twin Structure.